Infinite Series Volume Calculation

This problem involves finding the sum of an infinite geometric series. The volumes of the boxes form a sequence where each term is half of the previous one.

Identifying Series Parameters

• The first term (largest box volume), a = $20\text{ cc}$.
• The common ratio, r, is the factor by which each volume decreases. Since each box has half the volume of the previous one, r = $1/2$.

Sum of Infinite Geometric Series

The formula for the sum (S) of an infinite geometric series is:

$ S = \frac{a}{1-r} $

This formula is valid only if the absolute value of the common ratio is less than 1 (i.e., $|r| < 1$). In this case, $|1/2| < 1$, so the formula can be applied.

Calculating Total Volume

Substitute the values of a and r into the formula:

$ S = \frac{20}{1 - \frac{1}{2}} $

First, calculate the denominator:

$ 1 - \frac{1}{2} = \frac{1}{2} $

Now, perform the division:

$ S = \frac{20}{\frac{1}{2}} = 20 \times 2 = 40 $

The total volume of all the boxes is $40\text{ cc}$.